scaddins/source/analysis/bessel.cxx

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  19
  20 #include "bessel.hxx"
  21
  22 #include <rtl/math.hxx>
  23
  24 #include <com/sun/star/lang/IllegalArgumentException.hpp>
  25 #include <com/sun/star/sheet/NoConvergenceException.hpp>
  26
  27 using ::com::sun::star::lang::IllegalArgumentException;
  28 using ::com::sun::star::sheet::NoConvergenceException;
  29
  30 namespace sca {
  31 namespace analysis {
  32
  33 const double f_PI       = 3.1415926535897932385;
  34 const double f_PI_DIV_2 = f_PI / 2.0;
  35 const double f_PI_DIV_4 = f_PI / 4.0;
  36 const double f_2_DIV_PI = 2.0 / f_PI;
  37
  38
  39 // BESSEL J
  40
  41
  42 /*  The BESSEL function, first kind, unmodified:
  43     The algorithm follows
  44     http://www.reference-global.com/isbn/978-3-11-020354-7
  45     Numerical Mathematics 1 / Numerische Mathematik 1,
  46     An algorithm-based introduction / Eine algorithmisch orientierte Einfuehrung
  47     Deuflhard, Peter; Hohmann, Andreas
  48     Berlin, New York (Walter de Gruyter) 2008
  49     4. ueberarb. u. erw. Aufl. 2008
  50     eBook ISBN: 978-3-11-020355-4
  51     Chapter 6.3.2 , algorithm 6.24
  52     The source is in German.
  53     The BesselJ-function is a special case of the adjoint summation with
  54     a_k = 2*(k-1)/x for k=1,...
  55     b_k = -1, for all k, directly substituted
  56     m_0=1, m_k=2 for k even, and m_k=0 for k odd, calculated on the fly
  57     alpha_k=1 for k=N and alpha_k=0 otherwise
  58 */
  59
  60 double BesselJ( double x, sal_Int32 N )
  61
  62 {
  63     if( N < 0 )
  64         throw IllegalArgumentException();
  65     if (x==0.0)
  66         return (N==0) ? 1.0 : 0.0;
  67
  68     /*  The algorithm works only for x>0, therefore remember sign. BesselJ
  69         with integer order N is an even function for even N (means J(-x)=J(x))
  70         and an odd function for odd N (means J(-x)=-J(x)).*/
  71     double fSign = (N % 2 == 1 && x < 0) ? -1.0 : 1.0;
  72     double fX = fabs(x);
  73
  74     const double fMaxIteration = 9000000.0; //experimental, for to return in < 3 seconds
  75     double fEstimateIteration = fX * 1.5 + N;
  76     bool bAsymptoticPossible = pow(fX,0.4) > N;
  77     if (fEstimateIteration > fMaxIteration)
  78     {
  79         if (!bAsymptoticPossible)
  80             throw NoConvergenceException();
  81         return fSign * sqrt(f_2_DIV_PI/fX)* cos(fX-N*f_PI_DIV_2-f_PI_DIV_4);
  82     }
  83
  84     double const epsilon = 1.0e-15; // relative error
  85     bool bHasfound = false;
  86     double k= 0.0;
  87     // e_{-1} = 0; e_0 = alpha_0 / b_2
  88     double  u ; // u_0 = e_0/f_0 = alpha_0/m_0 = alpha_0
  89
  90     // first used with k=1
  91     double m_bar;         // m_bar_k = m_k * f_bar_{k-1}
  92     double g_bar;         // g_bar_k = m_bar_k - a_{k+1} + g_{k-1}
  93     double g_bar_delta_u; // g_bar_delta_u_k = f_bar_{k-1} * alpha_k
  94                           // - g_{k-1} * delta_u_{k-1} - m_bar_k * u_{k-1}
  95     // f_{-1} = 0.0; f_0 = m_0 / b_2 = 1/(-1) = -1
  96     double g = 0.0;       // g_0= f_{-1} / f_0 = 0/(-1) = 0
  97     double delta_u = 0.0; // dummy initialize, first used with * 0
  98     double f_bar = -1.0;  // f_bar_k = 1/f_k, but only used for k=0
  99
 100     if (N==0)
 101     {
 102         //k=0; alpha_0 = 1.0
 103         u = 1.0; // u_0 = alpha_0
 104         // k = 1.0; at least one step is necessary
 105         // m_bar_k = m_k * f_bar_{k-1} ==> m_bar_1 = 0.0
 106         g_bar_delta_u = 0.0;    // alpha_k = 0.0, m_bar = 0.0; g= 0.0
 107         g_bar = - 2.0/fX;       // k = 1.0, g = 0.0
 108         delta_u = g_bar_delta_u / g_bar;
 109         u = u + delta_u ;       // u_k = u_{k-1} + delta_u_k
 110         g = -1.0 / g_bar;       // g_k=b_{k+2}/g_bar_k
 111         f_bar = f_bar * g;      // f_bar_k = f_bar_{k-1}* g_k
 112         k = 2.0;
 113         // From now on all alpha_k = 0.0 and k > N+1
 114     }
 115     else
 116     {   // N >= 1 and alpha_k = 0.0 for k<N
 117         u=0.0; // u_0 = alpha_0
 118         for (k =1.0; k<= N-1; k = k + 1.0)
 119         {
 120             m_bar=2.0 * fmod(k-1.0, 2.0) * f_bar;
 121             g_bar_delta_u = - g * delta_u - m_bar * u; // alpha_k = 0.0
 122             g_bar = m_bar - 2.0*k/fX + g;
 123             delta_u = g_bar_delta_u / g_bar;
 124             u = u + delta_u;
 125             g = -1.0/g_bar;
 126             f_bar=f_bar * g;
 127         }
 128         // Step alpha_N = 1.0
 129         m_bar=2.0 * fmod(k-1.0, 2.0) * f_bar;
 130         g_bar_delta_u = f_bar - g * delta_u - m_bar * u; // alpha_k = 1.0
 131         g_bar = m_bar - 2.0*k/fX + g;
 132         delta_u = g_bar_delta_u / g_bar;
 133         u = u + delta_u;
 134         g = -1.0/g_bar;
 135         f_bar = f_bar * g;
 136         k = k + 1.0;
 137     }
 138     // Loop until desired accuracy, always alpha_k = 0.0
 139     do
 140     {
 141         m_bar = 2.0 * fmod(k-1.0, 2.0) * f_bar;
 142         g_bar_delta_u = - g * delta_u - m_bar * u;
 143         g_bar = m_bar - 2.0*k/fX + g;
 144         delta_u = g_bar_delta_u / g_bar;
 145         u = u + delta_u;
 146         g = -1.0/g_bar;
 147         f_bar = f_bar * g;
 148         bHasfound = (fabs(delta_u)<=fabs(u)*epsilon);
 149         k = k + 1.0;
 150     }
 151     while (!bHasfound && k <= fMaxIteration);
 152     if (!bHasfound)
 153         throw NoConvergenceException(); // unlikely to happen
 154
 155     return u * fSign;
 156 }
 157
 158
 159 // BESSEL I
 160
 161
 162 /*  The BESSEL function, first kind, modified:
 163
 164                      inf                                  (x/2)^(n+2k)
 165         I_n(x)  =  SUM   TERM(n,k)   with   TERM(n,k) := --------------
 166                      k=0                                   k! (n+k)!
 167
 168     No asymptotic approximation used, see issue 43040.
 169  */
 170
 171 double BesselI( double x, sal_Int32 n )
 172 {
 173     const sal_Int32 nMaxIteration = 2000;
 174     const double fXHalf = x / 2.0;
 175     if( n < 0 )
 176         throw IllegalArgumentException();
 177
 178     double fResult = 0.0;
 179
 180     /*  Start the iteration without TERM(n,0), which is set here.
 181
 182             TERM(n,0) = (x/2)^n / n!
 183      */
 184     sal_Int32 nK = 0;
 185     double fTerm = 1.0;
 186     // avoid overflow in Fak(n)
 187     for( nK = 1; nK <= n; ++nK )
 188     {
 189         fTerm = fTerm / static_cast< double >( nK ) * fXHalf;
 190     }
 191     fResult = fTerm;    // Start result with TERM(n,0).
 192     if( fTerm != 0.0 )
 193     {
 194         nK = 1;
 195         const double fEpsilon = 1.0E-15;
 196         do
 197         {
 198             /*  Calculation of TERM(n,k) from TERM(n,k-1):
 199
 200                                    (x/2)^(n+2k)
 201                     TERM(n,k)  =  --------------
 202                                     k! (n+k)!
 203
 204                                    (x/2)^2 (x/2)^(n+2(k-1))
 205                                =  --------------------------
 206                                    k (k-1)! (n+k) (n+k-1)!
 207
 208                                    (x/2)^2     (x/2)^(n+2(k-1))
 209                                =  --------- * ------------------
 210                                    k(n+k)      (k-1)! (n+k-1)!
 211
 212                                    x^2/4
 213                                =  -------- TERM(n,k-1)
 214                                    k(n+k)
 215             */
 216         fTerm = fTerm * fXHalf / static_cast<double>(nK) * fXHalf / static_cast<double>(nK+n);
 217         fResult += fTerm;
 218         nK++;
 219         }
 220         while( (fabs( fTerm ) > fabs(fResult) * fEpsilon) && (nK < nMaxIteration) );
 221
 222     }
 223     return fResult;
 224 }
 225
 226 /// @throws IllegalArgumentException
 227 /// @throws NoConvergenceException
 228 static double Besselk0( double fNum )
 229 {
 230     double  fRet;
 231
 232     if( fNum <= 2.0 )
 233     {
 234         double  fNum2 = fNum * 0.5;
 235         double  y = fNum2 * fNum2;
 236
 237         fRet = -log( fNum2 ) * BesselI( fNum, 0 ) +
 238                 ( -0.57721566 + y * ( 0.42278420 + y * ( 0.23069756 + y * ( 0.3488590e-1 +
 239                     y * ( 0.262698e-2 + y * ( 0.10750e-3 + y * 0.74e-5 ) ) ) ) ) );
 240     }
 241     else
 242     {
 243         double  y = 2.0 / fNum;
 244
 245         fRet = exp( -fNum ) / sqrt( fNum ) * ( 1.25331414 + y * ( -0.7832358e-1 +
 246                 y * ( 0.2189568e-1 + y * ( -0.1062446e-1 + y * ( 0.587872e-2 +
 247                 y * ( -0.251540e-2 + y * 0.53208e-3 ) ) ) ) ) );
 248     }
 249
 250     return fRet;
 251 }
 252
 253 /// @throws IllegalArgumentException
 254 /// @throws NoConvergenceException
 255 static double Besselk1( double fNum )
 256 {
 257     double  fRet;
 258
 259     if( fNum <= 2.0 )
 260     {
 261         double  fNum2 = fNum * 0.5;
 262         double  y = fNum2 * fNum2;
 263
 264         fRet = log( fNum2 ) * BesselI( fNum, 1 ) +
 265                 ( 1.0 + y * ( 0.15443144 + y * ( -0.67278579 + y * ( -0.18156897 + y * ( -0.1919402e-1 +
 266                     y * ( -0.110404e-2 + y * -0.4686e-4 ) ) ) ) ) )
 267                 / fNum;
 268     }
 269     else
 270     {
 271         double  y = 2.0 / fNum;
 272
 273         fRet = exp( -fNum ) / sqrt( fNum ) * ( 1.25331414 + y * ( 0.23498619 +
 274                 y * ( -0.3655620e-1 + y * ( 0.1504268e-1 + y * ( -0.780353e-2 +
 275                 y * ( 0.325614e-2 + y * -0.68245e-3 ) ) ) ) ) );
 276     }
 277
 278     return fRet;
 279 }
 280
 281
 282 double BesselK( double fNum, sal_Int32 nOrder )
 283 {
 284     switch( nOrder )
 285     {
 286         case 0:     return Besselk0( fNum );
 287         case 1:     return Besselk1( fNum );
 288         default:
 289         {
 290             double      fTox = 2.0 / fNum;
 291             double      fBkm = Besselk0( fNum );
 292             double      fBk = Besselk1( fNum );
 293
 294             for( sal_Int32 n = 1 ; n < nOrder ; n++ )
 295             {
 296                 const double fBkp = fBkm + double( n ) * fTox * fBk;
 297                 fBkm = fBk;
 298                 fBk = fBkp;
 299             }
 300
 301             return fBk;
 302         }
 303     }
 304 }
 305
 306
 307 // BESSEL Y
 308
 309
 310 /*  The BESSEL function, second kind, unmodified:
 311     The algorithm for order 0 and for order 1 follows
 312     http://www.reference-global.com/isbn/978-3-11-020354-7
 313     Numerical Mathematics 1 / Numerische Mathematik 1,
 314     An algorithm-based introduction / Eine algorithmisch orientierte Einfuehrung
 315     Deuflhard, Peter; Hohmann, Andreas
 316     Berlin, New York (Walter de Gruyter) 2008
 317     4. ueberarb. u. erw. Aufl. 2008
 318     eBook ISBN: 978-3-11-020355-4
 319     Chapter 6.3.2 , algorithm 6.24
 320     The source is in German.
 321     See #i31656# for a commented version of the implementation, attachment #desc6
 322     https://bz.apache.org/ooo/attachment.cgi?id=63609
 323 */
 324
 325 /// @throws IllegalArgumentException
 326 /// @throws NoConvergenceException
 327 static double Bessely0( double fX )
 328 {
 329     if (fX <= 0)
 330         throw IllegalArgumentException();
 331     const double fMaxIteration = 9000000.0; // should not be reached
 332     if (fX > 5.0e+6) // iteration is not considerable better then approximation
 333         return sqrt(1/f_PI/fX)
 334                 *(rtl::math::sin(fX)-rtl::math::cos(fX));
 335     const double epsilon = 1.0e-15;
 336     const double EulerGamma = 0.57721566490153286060;
 337     double alpha = log(fX/2.0)+EulerGamma;
 338     double u = alpha;
 339
 340     double k = 1.0;
 341     double g_bar_delta_u = 0.0;
 342     double g_bar = -2.0 / fX;
 343     double delta_u = g_bar_delta_u / g_bar;
 344     double g = -1.0/g_bar;
 345     double f_bar = -1 * g;
 346
 347     double sign_alpha = 1.0;
 348     bool bHasFound = false;
 349     k = k + 1;
 350     do
 351     {
 352         double km1mod2 = fmod(k-1.0, 2.0);
 353         double m_bar = (2.0*km1mod2) * f_bar;
 354         if (km1mod2 == 0.0)
 355             alpha = 0.0;
 356         else
 357         {
 358            alpha = sign_alpha * (4.0/k);
 359            sign_alpha = -sign_alpha;
 360         }
 361         g_bar_delta_u = f_bar * alpha - g * delta_u - m_bar * u;
 362         g_bar = m_bar - (2.0*k)/fX + g;
 363         delta_u = g_bar_delta_u / g_bar;
 364         u = u+delta_u;
 365         g = -1.0 / g_bar;
 366         f_bar = f_bar*g;
 367         bHasFound = (fabs(delta_u)<=fabs(u)*epsilon);
 368         k=k+1;
 369     }
 370     while (!bHasFound && k<fMaxIteration);
 371     if (!bHasFound)
 372         throw NoConvergenceException(); // not likely to happen
 373     return u*f_2_DIV_PI;
 374 }
 375
 376 // See #i31656# for a commented version of this implementation, attachment #desc6
 377 // https://bz.apache.org/ooo/attachment.cgi?id=63609
 378 /// @throws IllegalArgumentException
 379 /// @throws NoConvergenceException
 380 static double Bessely1( double fX )
 381 {
 382     if (fX <= 0)
 383         throw IllegalArgumentException();
 384     const double fMaxIteration = 9000000.0; // should not be reached
 385     if (fX > 5.0e+6) // iteration is not considerable better then approximation
 386         return - sqrt(1/f_PI/fX)
 387                 *(rtl::math::sin(fX)+rtl::math::cos(fX));
 388     const double epsilon = 1.0e-15;
 389     const double EulerGamma = 0.57721566490153286060;
 390     double alpha = 1.0/fX;
 391     double f_bar = -1.0;
 392     double u = alpha;
 393     double k = 1.0;
 394     alpha = 1.0 - EulerGamma - log(fX/2.0);
 395     double g_bar_delta_u = -alpha;
 396     double g_bar = -2.0 / fX;
 397     double delta_u = g_bar_delta_u / g_bar;
 398     u = u + delta_u;
 399     double g = -1.0/g_bar;
 400     f_bar = f_bar * g;
 401     double sign_alpha = -1.0;
 402     bool bHasFound = false;
 403     k = k + 1.0;
 404     do
 405     {
 406         double km1mod2 = fmod(k-1.0,2.0);
 407         double m_bar = (2.0*km1mod2) * f_bar;
 408         double q = (k-1.0)/2.0;
 409         if (km1mod2 == 0.0) // k is odd
 410         {
 411            alpha = sign_alpha * (1.0/q + 1.0/(q+1.0));
 412            sign_alpha = -sign_alpha;
 413         }
 414         else
 415             alpha = 0.0;
 416         g_bar_delta_u = f_bar * alpha - g * delta_u - m_bar * u;
 417         g_bar = m_bar - (2.0*k)/fX + g;
 418         delta_u = g_bar_delta_u / g_bar;
 419         u = u+delta_u;
 420         g = -1.0 / g_bar;
 421         f_bar = f_bar*g;
 422         bHasFound = (fabs(delta_u)<=fabs(u)*epsilon);
 423         k=k+1;
 424     }
 425     while (!bHasFound && k<fMaxIteration);
 426     if (!bHasFound)
 427         throw NoConvergenceException();
 428     return -u*2.0/f_PI;
 429 }
 430
 431 double BesselY( double fNum, sal_Int32 nOrder )
 432 {
 433     switch( nOrder )
 434     {
 435         case 0:     return Bessely0( fNum );
 436         case 1:     return Bessely1( fNum );
 437         default:
 438         {
 439             double      fTox = 2.0 / fNum;
 440             double      fBym = Bessely0( fNum );
 441             double      fBy = Bessely1( fNum );
 442
 443             for( sal_Int32 n = 1 ; n < nOrder ; n++ )
 444             {
 445                 const double fByp = double( n ) * fTox * fBy - fBym;
 446                 fBym = fBy;
 447                 fBy = fByp;
 448             }
 449
 450             return fBy;
 451         }
 452     }
 453 }
 454
 455 } // namespace analysis
 456 } // namespace sca
 457
 458 /* vim:set shiftwidth=4 softtabstop=4 expandtab: */